The derived series of a group $G$ is the descending normal series of $G$ whose terms are the successive derived subgroups, defined as follows. Let $G_0=G$ and, for $0<k$, define $G_k=\left[G_{k-1}\&comma;G_{k-1}\right]$. The sequence

of distinct terms is called the derived series of $G$. The number $r$ is called the derived length of $G$, and the soluble residual $G_r$ of $G$ is the last term of the derived series. If the soluble residual $G_r$ is the trivial group, then we say that $G$ is soluble (or solvable).

The DerivedSeries( G ) command constructs the derived series of a group G. The group G must be an instance of a permutation group. The derived series of G is represented by a series data structure which admits certain operations common to all series.  See GroupTheory[Series].

The DerivedLength( G ) command returns the derived length of G; that is, the length of the derived series of G. This is the number of subgroup inclusions - so it is one less than the number of groups in the derived series.

The IsSoluble( G ) command (or IsSolvable( G ), as an alias) returns true if the group G is soluble, and returns the value false otherwise.

The SolubleResidual( G ) command (or SolvableResidual( G ), as an alias) returns the soluble residual $G_r$ of $G$. It can also be applied to a derived series object.

The lower Fitting series of a group $G$ is the descending normal series of $G$ whose terms are the successive nilpotent residuals. The sequence

of distinct terms is called the lower Fitting series of $G$ if $G_{i+1}$ is defined to be the nilpotent residual of $G_i$. Then $G$ is soluble if, and only if, this series reaches the trivial subgroup. Its length is called the Fitting length (also known as the nilpotent length) of $G$.

Note that the Fitting length of $G$ is equal to $1$ precisely when $G$ is nilpotent as, in this case, the nilpotent residual of $G$ is trivial.

The LowerFittingSeries( G ) command computes the lower Fitting series of the permutation group G. Like the derived series, the lower Fitting series of G is represented by a series data structure.  Again, see GroupTheory[Series].

The FittingLength( G ) returns the Fitting length of G.  The NilpotentLength( G ) command is identical, and is provided as an alias.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\&colon;$

$G≔\mathrm{PermutationGroup}\left(\left\{\left[\left[1\&comma;2\right]\right]\&comma;\left[\left[1\&comma;2\&comma;3\right]\&comma;\left[4\&comma;5\right]\right]\right\}\right)$

$G≔⟨\left(1\&comma;2\right)\&comma;\left(1\&comma;2\&comma;3\right)\left(4\&comma;5\right)⟩$

$\mathrm{ds}≔\mathrm{DerivedSeries}\left(G\right)$

$\mathrm{ds}≔⟨\left(1\&comma;2\right)\&comma;\left(1\&comma;2\&comma;3\right)\left(4\&comma;5\right)⟩▹\left[⟨\left(1\&comma;2\right)\&comma;\left(1\&comma;2\&comma;3\right)\left(4\&comma;5\right)⟩\&comma;⟨\left(1\&comma;2\right)\&comma;\left(1\&comma;2\&comma;3\right)\left(4\&comma;5\right)⟩\right]▹⟨⟩$

$\mathrm{DerivedLength}\left(G\right)$

$2$

$\mathrm{IsSoluble}\left(G\right)$

$\mathrm{true}$

$\mathrm{type}\left(\mathrm{ds}\&comma;'\mathrm{NormalSeries}'\right)$

$\mathrm{true}$

$\mathrm{ds}≔\mathrm{DerivedSeries}\left(\mathrm{AlternatingGroup}\left(4\right)\right)$

$\mathrm{ds}≔{\mathbf{A}}_4▹\left[{\mathbf{A}}_4\&comma;{\mathbf{A}}_4\right]▹⟨⟩$

$$\begin{gathered} \mathbf{for}g\mathbf{in}\mathrm{ds}\mathbf{do} \\ \mathrm{print}\left(g\right) \\ \mathbf{end}\mathbf{do}\&colon; \end{gathered}$$

${\mathbf{A}}_4$

$\left[{\mathbf{A}}_4\&comma;{\mathbf{A}}_4\right]$

$\mathrm{SolubleResidual}\left(\mathrm{ds}\right)$

$\mathrm{LowerFittingSeries}\left(\mathrm{Alt}\left(4\right)\right)$

${\mathbf{A}}_4▹\left[{\mathbf{A}}_4\&comma;{\mathbf{A}}_4\right]▹⟨⟩$

$\mathrm{FittingLength}\left(\mathrm{Symm}\left(4\right)\right)$

$3$

$\mathrm{FittingLength}\left(\mathrm{DihedralGroup}\left(8\right)\right)$

$1$

The GroupTheory[DerivedSeries], GroupTheory[DerivedLength], GroupTheory[IsSoluble] and GroupTheory[SolubleResidual] commands were introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.

The GroupTheory[LowerFittingSeries], GroupTheory[FittingLength] and GroupTheory[NilpotentLength] commands were introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.